Formal Normal Forms for the Perturbations of a Quasi-Homogeneous Hamiltonian Vector Field

Abstract

We classify up to a formal change of variable the vector fields of two complex variables together with the foliations that they de.ne which are a perturbation of a quasi-homogeneous Hamiltonian vector field X0 by terms of higher degree of quasi-homogeneity. We do not require anything about the degree δ0 of the initial vector field X0, but we assume that the perturbed vector field still keeps invariant the separatrices of X0. We obtain formal normal forms which extend those obtained in the case of an initial vector field with a semisimple or nilpotent linear part. We give an interpretation of the dual version of these normal forms through the relative cohomology with respect to the dual initial di.erential form ω0.